I am considering purchasing a mount in the DB store with saved up DBC. If I do, does it only work for the character I am currently logged in as when I get it, or can I use it on multiple characters?
You cannot pass it to other accounts, but if the item itself is marked as Heirloom, that means you can pass it to other characters on the same account/server. The item attributes are right under its name when you examine the stats of an item - Heirloom would be with other descriptors like No Trade, Lore, etc.
Trying to understand your question, I will say that if you buy a mount you can only use it on one character. You may, if it is Heirloom, give it to another character on the same account and the same game server. But you only get one mount which can only be used on one character.